Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
aa r X i v : . [ m a t h . A P ] M a r Two classes of nonlocal Evolution Equations related by ashared Traveling Wave Problem
Franz Achleitner ∗ July 23, 2018
Abstract
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations,i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existenceof traveling wave solutions for these two classes of evolution equations. For classical equationsthe traveling wave problem (TWP) for a local KdVB equation can be identified with the TWPfor a reaction-diffusion equation. In this article we study this relationship for these two classesof evolution equations with nonlocal diffusion/dispersion. This connection is especially useful,if the TW equation is not studied directly, but the existence of a TWS is proven using oneof the evolution equations instead. Finally, we present three models from fluid dynamics anddiscuss the TWP via its link to associated reaction-diffusion equations.
We will consider two classes of (nonlocal) evolution equations and study the associated trav-eling wave problems in parallel: On the one hand, we consider scalar conservation laws with(nonlocal) diffusive-dispersive regularization ∂ t u + ∂ x f ( u ) = ǫ L [ u ] + δ∂ x L [ u ] , t > , x ∈ R , (1)for some nonlinear function f : R → R , L´evy operators L and L , as well as constants ǫ, δ ∈ R .The Fourier multiplier operators L and ∂ x L model diffusion and dispersion, respectively. Onthe other hand, we consider scalar reaction-diffusion equations ∂ t u = r ( u ) + σ L [ u ] , t > , x ∈ R , (2)for some positive constant σ , as well as a nonlinear function r : R → R and a L´evy operator L modeling reaction and diffusion, respectively. Definition 1.
A traveling wave solution (TWS) of an evolution equation–such as (1) and (2)–is a solution u ( x, t ) = ¯ u ( ξ ) whose profile ¯ u depends on ξ := x − ct for some wave speed c .Moreover, the profile ¯ u ∈ C ( R ) is assumed to approach distinct endstates u ± such thatlim ξ →±∞ ¯ u ( ξ ) = u ± , lim ξ →±∞ ¯ u ( n ) ( ξ ) = 0 with n = 1 ,
2. (3) ∗ TU Wien, Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria,[email protected] uch a TWS is also known as a front in the literature. A TWS (¯ u, c ) is called monotone, ifits profile ¯ u is a monotone function. Definition 2.
The traveling wave problem (TWP) associated to an evolution equation is tostudy for some distinct endstates u ± the existence of a TWS (¯ u, c ) in the sense of Definition 1.We want to identify classes of evolutions equations of type (1) and (2), which lead to thesame TWP. This connection is especially useful, if the TWP is not studied directly, but theexistence of a TWS is proven using one of the evolution equations instead. A classical exampleof (1) is a scalar conservation law with local diffusive-dispersive regularization ∂ t u + ∂ x f ( u ) = ǫ∂ x u + δ∂ x u , t > , x ∈ R , (4)for some nonlinear function f : R → R and some constants ǫ > δ ∈ R . Equation (4) withBurgers flux f ( u ) = u is known as Korteweg-de Vries-Burgers (KdVB) equation; hence werefer to Equation (4) with general f as generalized KdVB equation and Equation (1) as nonlocalgeneralized
KdVB equation. A TWS (¯ u, c ) satisfies the traveling wave equation (TWE) − c ¯ u ′ + f ′ (¯ u ) ¯ u ′ = ǫ ¯ u ′′ + δ ¯ u ′′′ , ξ ∈ R , (5)or integrating on ( −∞ , ξ ] and using (3), h (¯ u ) := f (¯ u ) − c ¯ u − ( f ( u − ) − c u − ) = ǫ ¯ u ′ + δ ¯ u ′′ , ξ ∈ R . (6)However, the TW ansatz v ( x, t ) = ¯ u ( x − ǫt ) for the scalar reaction-diffusion equation ∂ t v = − h ( v ) + δ∂ x v , t > , x ∈ R , (7)leads to the same TWE (6) except for a different interpretation of the parameters. The travelingwave speeds in the TWP of (4) and (7) are c and ǫ , respectively. For fixed parameters c , ǫ , and δ , the existence of a traveling wave profile ¯ u satisfying (3) and (6) reduces to the existence ofa heteroclinic orbit for this ODE. This is an example, where the existence of TWS is studieddirectly via the TWE.A first example, where the TWE is not studied directly, is the TWP for a nonlocal KdVBequation (1) with L [ u ] = ∂ x u and L [ u ] = φ ǫ ∗ u − u for some even non-negative function φ ∈ L ( R ) with compact support and unit mass, where φ ǫ ( · ) := φ ( · /ǫ ) /ǫ with ǫ >
0. It has beenderived as a model for phase transitions with long range interactions close to the surface, whichsupports planar TWS associated to undercompressive shocks of (51), see [41]. In particular,the TWP for a cubic flux function f ( u ) = u is related to the TWP for a reaction-diffusionequation (2) with L [ u ] = L [ u ]. The existence of TWS for this reaction-diffusion equation hasbeen proven via a homotopy of (2) to a classical reaction-diffusion model (7), see [11]. Outline.
In Section 2 we collect background material on L´evy operators L , which willmodel diffusion in our nonlocal evolution equations. Special emphasize is given to convolutionoperators and Riesz-Feller operators. In Section 3 we review the classical results on the TWP forreaction-diffusion equations (7) and generalized Korteweg-de Vries-Burgers equation (4). Westudy their relationship in detail, especially the classification of function h ( u ), which will beused again in Section 4. In Section 4, first we review the results on TWP for nonlocal reaction-diffusion equations (2) with operators L of convolution type and Riesz-Feller type, respectively.Finally, we study the example of nonlocal generalized Korteweg-de Vries-Burgers equationmodeling a shallow water flow [33] and Fowler’s equation modeling dune formation [26], ∂ t u + ∂ x u = ∂ x u − ∂ x D / u , t > , x ∈ R , (8)where D α + is a Caputo derivative. In the Appendix, we collect background material on Caputoderivatives D α + and the shock wave theory for scalar conservation laws, which will explain theimportance of the TWP for KdVB equations. otations. We use the conventions in probability theory, and define the Fourier transform F and its inverse F − for g ∈ L ( R ) and x, k ∈ R as F [ g ]( k ) := Z R e + i kx g ( x ) d x ; F − [ g ]( x ) := π Z R e − i kx g ( k ) d k . In the following, F and F − will denote also their respective extensions to L ( R ). A L´evy process is a stochastic process with independent and stationary increments which iscontinuous in probability [9, 30, 42]. Therefore a L´evy process is characterized by its transitionprobabilities p ( t, x ), which evolve according to an evolution equation ∂ t p = L p (9)for some operator L , also called a L´evy operator . First, we define L´evy operators on the functionspaces C ( R ) := { f ∈ C ( R ) | lim | x |→∞ f ( x ) = 0 } and C ( R ) := { f, f ′ , f ′′ ∈ C ( R ) } . Definition 3.
The family of L´evy operators in one spatial dimension consists of operators L defined for f ∈ C ( R ) as L f ( x ) = Af ′′ ( x ) + γ f ′ ( x ) + Z R (cid:0) f ( x + y ) − f ( x ) − y f ′ ( x )1 ( − , ( y ) (cid:1) ν ( d y ) (10)for some constants A ≥ γ ∈ R , and a measure ν on R satisfying ν ( { } ) = 0 and Z R min(1 , | y | ) ν ( d y ) < ∞ . Remark . The function f ( x + y ) − f ( x ) − y f ′ ( x )1 ( − , ( y ) is integrable with respect to ν ,because it is bounded outside of any neighborhood of 0 and f ( x + y ) − f ( x ) − y f ′ ( x )1 ( − , ( y ) = O ( | y | ) as | y | → x . The indicator function c ( y ) = 1 ( − , ( y ) is only one possible choice to obtain anintegrable integrand. More generally, let c ( y ) be a bounded measurable function from R to R satisfying c ( y ) = 1 + o ( | y | ) as | y | →
0, and c ( y ) = O (1 / | y | ) as | y | → ∞ . Then (10) is rewrittenas L f ( x ) = Af ′′ ( x ) + γ c f ′ ( x ) + Z R (cid:0) f ( x + y ) − f ( x ) − y f ′ ( x ) c ( y ) (cid:1) ν ( d y ) , (11)with γ c = γ + R R y ( c ( y ) − ( − , ( y )) ν ( d y ) .Alternative choices for c :(c 0) If a L´evy measure ν satisfies R | y | < | y | ν ( d y ) < ∞ then c ≡ ν satisfies R | y | > | y | ν ( d y ) < ∞ then c ≡ A and ν are invariant no matter what function c we choose. Examples. d − Id ¶ x − ¶ x − D a + u D a + u ¶ x D a + u ¶ x ¶ x − a + a a q − Figure 1: The family of Fourier multipliers ψ aθ ( k ) = −| k | a exp (cid:2) i sgn( k ) θπ/ (cid:3) has two parameters a and θ . Some Fourier multiplier operators F [ T f ]( k ) = ψ aθ ( k ) F [ f ]( k ) are inserted in the parameterspace ( a, θ ): partial derivatives and Caputo derivatives D α + with 0 < α <
1. The Riesz-Felleroperators D aθ are those operators with parameters ( a, θ ) ∈ D a,θ . The set D a,θ is also called Feller-Takayasu diamond and depicted as a shaded region, see also [36]. (a) The L´evy operators L f = Z R (cid:0) f ( x + y ) − f ( x ) (cid:1) ν ( d y ) (12)are infinitesimal generators associated to a compound Poisson process with finite L´evymeasure ν satisfying (c 0). The special case of ν ( d y ) = φ ( − y ) d y for some function φ ∈ L ( R ) yields L f ( x ) = Z R (cid:0) f ( x + y ) − f ( x ) (cid:1) φ ( − y ) d y = (cid:0) φ ∗ f − Z R φ d y f (cid:1) ( x ) . (13)(b) Riesz-Feller operators.
The Riesz-Feller operators of order a and asymmetry θ are definedas Fourier multiplier operators F [ D aθ f ]( k ) = ψ aθ ( k ) F [ f ]( k ) , k ∈ R , (14)with symbol ψ aθ ( k ) = −| k | a exp [i sgn( k ) θπ/
2] such that ( a, θ ) ∈ D a,θ and D a,θ := { ( a, θ ) ∈ R | < a ≤ , | θ | ≤ min { a, − a } } . Special cases of Riesz-Feller operators are • Fractional Laplacians − ( − ∆) a/ on R with Fourier symbol −| k | a for 0 < a ≤
2. Inparticular, fractional Laplacians are the only symmetric Riesz-Feller operators with − ( − ∆) a/ = D a and θ ≡ • Caputo derivatives −D α + with 0 < α < a = α and θ = − α , such that −D α + = D α − α , see also Section A. • Derivatives of Caputo derivatives ∂ x D α + with 0 < α < a = 1 + α and θ = 1 − α , such that ∂ x D α + = D α − α . ext we consider the Cauchy problem ∂ t u ( x, t ) = D aθ [ u ( · , t )]( x ) , u ( x,
0) = u ( x ) , (15)for ( x, t ) ∈ R × (0 , ∞ ) and initial datum u . Proposition 1.
For ( a, θ ) ∈ D a,θ and ≤ p < ∞ , the Riesz-Feller operator D aθ generates astrongly continuous L p -semigroup S t : L p ( R ) → L p ( R ) , u S t u = G aθ ( · , t ) ∗ u , with heat kernel G aθ ( x, t ) = F − [exp( t ψ aθ ( · ))]( x ) . In particular, G aθ ( x, t ) is the probabilitymeasure of a L´evy strictly a -stable distribution. For ( a, θ ) ∈ { (1 , , (1 , − } , the probability measure G aθ is a delta distribution, e.g. G ( x, t ) = δ x + t and G − ( x, t ) = δ x − t , and is called trivial [42, Definition 13.6]. However, we are interestedin non-trivial probability measures G aθ for( a, θ ) ∈ D ⋄ a,θ := { ( a, θ ) ∈ D a,θ | | θ | < } , such that D a,θ = D ⋄ a,θ ∪ { (1 , , (1 , − } . Note, nonlocal Riesz-Feller D aθ operators are thosewith parameters ( a, θ ) ∈ D • a,θ := { ( a, θ ) ∈ D a,θ | < a < , | θ | < } , such that D ⋄ a,θ = D • a,θ ∪ { (2 , } . Proposition 2 ([6, Lemma 2.1]) . For ( a, θ ) ∈ D ⋄ a,θ the probability measure G aθ is absolutelycontinuous with respect to the Lebesgue measure and possesses a probability density which willbe denoted again by G aθ . For all ( x, t ) ∈ R × (0 , ∞ ) the following properties hold;(a) G aθ ( x, t ) ≥ . If θ = ± a then G aθ ( x, t ) > ;(b) k G aθ ( · , t ) k L ( R ) = 1 ;(c) G aθ ( x, t ) = t − /a G aθ ( xt − /a , ;(d) G aθ ( · , s ) ∗ G aθ ( · , t ) = G aθ ( · , s + t ) for all s, t ∈ (0 , ∞ ) ;(e) G aθ ∈ C ∞ ( R × (0 , ∞ )) . The L´evy measure ν of a Riesz-Feller operator D aθ with ( a, θ ) ∈ D • a,θ is absolutely continuouswith respect to Lebesgue measure and satisfies ν ( d y ) = ( c − ( θ ) y − − a d y on (0 , ∞ ) ,c + ( θ ) | y | − − a d y on ( −∞ , , (16)with c ± ( θ ) = Γ(1 + a ) sin(( a ± θ ) π/ /π , see [36, 43].To study a TWP for evolution equations involving Riesz-Feller operators, it is necessary toextend the Riesz-Feller operators to C b ( R ). Their singular integral representations (10) maybe used to accomplish this task. Theorem 1 ([6]) . If ( a, θ ) ∈ D • a,θ with a = 1 , then for all f ∈ S ( R ) and x ∈ R D aθ f ( x ) = c + ( θ ) − c − ( θ )1 − a f ′ ( x )+ c + ( θ ) Z ∞ f ( x + y ) − f ( x ) − f ′ ( x ) y ( − , ( y ) y a d y (17)+ c − ( θ ) Z ∞ f ( x − y ) − f ( x ) + f ′ ( x ) y ( − , ( y ) y a d y with c ± ( θ ) = Γ(1 + a ) sin(( a ± θ ) π/ /π . Alternative representations are If < a < , then D aθ f ( x ) = c + ( θ ) Z ∞ f ( x + y ) − f ( x ) y a d y + c − ( θ ) Z ∞ f ( x − y ) − f ( x ) y a d y . • If < a < , then D aθ f ( x ) = c + ( θ ) Z ∞ f ( x + y ) − f ( x ) − f ′ ( x ) yy a d y + c − ( θ ) Z ∞ f ( x − y ) − f ( x ) + f ′ ( x ) yy a d y . (18)These representations allow to extend Riesz-Feller operators D aθ to C b ( R ) such that D aθ C b ( R ) ⊂ C b ( R ). For example, one can show Proposition 3 ([6, Proposition 2.4]) . For ( a, θ ) ∈ D a,θ with < a < , the integral represen-tation (18) of D aθ is well-defined for functions f ∈ C b ( R ) with sup x ∈ R | D aθ f ( x ) | ≤ Kk f ′′ k C b ( R ) M − a − a + 4 Kk f ′ k C b ( R ) M − a a − < ∞ (19) for some positive constants M and K = Γ(1+ a ) π | sin(( a + θ ) π ) + sin(( a − θ ) π ) | . Estimate (19) is a key estimate, which is used to adapt Chen’s approach [19] to the TWPfor nonlocal reaction-diffusion equations with Riesz-Feller operators [6].
In this section we review the importance of the TWP for reaction-diffusion equations and scalarconservation laws with higher-order regularizations, respectively.
A scalar reaction-diffusion equations is a partial differential equation ∂ t u = r ( u ) + σ∂ x u , t > , x ∈ R , (20)for some positive constant σ >
0, as well as a nonlinear function r : R → R and second-orderderivative ∂ x u modeling reaction and diffusion, respectively. The TWP for given endstates u ± is to study the existence of a TWS (¯ u, c ) for (20) in the sense of Definition 1. If the profile¯ u ∈ C ( R ) is bounded, then it satisfies lim ξ →±∞ ¯ u ( n ) ( ξ ) = 0 for n = 1 ,
2. A TWS (¯ u, c ) satisfiesthe TWE − c ¯ u ′ = r (¯ u ) + σ ¯ u ′′ , ξ ∈ R . (21) phase plane analysis. A traveling wave profile ¯ u is a heteroclinic orbit of the TWE (21) con-necting the endstates u ± . To identify necessary conditions on the existence of TWS, TWE (21)is written as a system of first-order ODEs for u , v := u ′ :dd ξ uv ! = v ( − r ( u ) − cv ) /σ ! =: F ( u, v ) , ξ ∈ R . (22)First, an endstate ( u s , v s ) of a heteroclinic orbit has to be a stationary state of F , i.e. F ( u s , v s ) =0, which implies v s ≡ r ( u s ) = 0. Second, ( u − ,
0) has to be an unstable stationary state f (22) and ( u + ,
0) either a saddle or a stable node of (22). As long as a stationary state( u s , v s ) is hyperbolic, i.e. the linearization of F at ( u s , v s ) has only eigenvalues λ with non-zeroreal part, the stability of ( u s , v s ) is determined by these eigenvalues. The linearization of F at( u s , v s ) is D F ( u s , v s ) = (cid:18) − r ′ ( u s ) /σ − c/σ (cid:19) . (23)Eigenvalues λ ± of the Jacobian D F ( u s , v s ) satisfy the characteristic equation λ + λc/σ + r ′ ( u s ) /σ = 0. Moreover, λ − + λ + = − c/σ and λ − λ + = r ′ ( u s ) /σ . The eigenvalues λ ± of theJacobian D F ( u s , v s ) are λ ± = − c σ ± r c σ − r ′ ( u s ) σ = − c ± p c − σr ′ ( u s )2 σ . (24)Thus r ′ ( u s ) < u s ,
0) is a saddle point, i.e. with one positive and one negativeeigenvalue. balance of potential.
The potential R (of the reaction term r ) is defined as R ( u ) := R u r ( υ ) d υ . The potentials of the endstates u ± are called balanced if R ( u + ) = R ( u − )and unbalanced otherwise. A formal computation reveals a connection between the sign of c and the balance of the potential R ( u ): Multiplying TWE (21) with ¯ u ′ , integrating on R andusing (3), yields − c k ¯ u ′ k L = Z u + u − r ( υ ) d υ = R ( u + ) − R ( u − ) , (25)since R R ¯ u ′′ ¯ u ′ d ξ = 0 due to (3). Thus − sgn c = sgn( R ( u + ) − R ( u − )). In case of a balancedpotential the wave speed c is zero, hence the TWS is stationary. Definition 4.
Assume u − > u + . A function r ∈ C ( R ) with r ( u ± ) = 0 is • monostable if r ′ ( u − ) < r ′ ( u + ) > r ( u ) > u ∈ ( u + , u − ). • bistable if r ′ ( u ± ) < ∃ u ∗ ∈ ( u + , u − ) : r ( u ) ( < u ∈ ( u + , u ∗ ) ,> u ∈ ( u ∗ , u − ) . • unstable if r ′ ( u ± ) > u models a density of a substance/population. In thesesituations only nonnegative states u ± and functions u are of interest. Proposition 4 ([45, § . Assume σ > and u − > u + . • If r is monostable, then there exists a positive constant c ∗ such that for all c ≥ c ∗ thereexists a monotone TWS (¯ u, c ) of (20) in the sense of Definition 1. For c < c ∗ no suchmonotone TWS exists (however oscillatory TWS may exist). • If r is bistable, then there exists an (up to translations) unique monotone TWS (¯ u, c ) of (20) in the sense of Definition 1. • If r is unstable, then there does not exist a monotone TWS (¯ u, c ) of (20) . If a TWS (¯ u, c ) exists, then a closer inspection of the eigenvalues (24) at ( u + ,
0) indicatesthe geometry of the profile ¯ u for large ξ : c − σr ′ ( u + ) ( ≥ u for large ξ ; < u for large ξ. .2 Korteweg-de Vries-Burgers equation (KdVB) A generalized KdVB equation is a scalar partial differential equation ∂ t u + ∂ x f ( u ) = ǫ∂ x u + δ∂ x u, x ∈ R , t > , (26)for some flux function f : R → R as well as constants ǫ > δ ∈ R . The TWP for givenendstates u ± is to study the existence of a TWS (¯ u, c ) for (26) in the sense of Definition 1. Theimportance of the TWP for KdVB equations in the shock wave theory of (scalar) hyperbolicconservation laws is discussed in Section B. A TWS (¯ u, c ) satisfies the TWE − c ¯ u ′ + f ′ (¯ u ) ¯ u ′ = ǫ ¯ u ′′ + δ ¯ u ′′′ , ξ ∈ R , (27)or integrating on ( −∞ , ξ ] and using (3), h (¯ u ) := f (¯ u ) − c ¯ u − ( f ( u − ) − c u − ) = ǫ ¯ u ′ + δ ¯ u ′′ , ξ ∈ R . (28) connection with reaction-diffusion equation. A TWS u ( x, t ) = ¯ u ( x − ct ) of a generalizedKorteweg-de Vries-Burgers equation (26) satisfies TWE (28). Thus v ( x, t ) = ¯ u ( x − ǫt ) is aTWS (¯ u, ǫ ) of the reaction-diffusion equation ∂ t v = − h ( v ) + δ∂ x v , x ∈ R , t > . (29) phase plane analysis. Following the analysis of TWE (21) for a reaction-diffusion equa-tion (20) with r ( u ) = − h ( u ) and σ = δ , necessary conditions on the parameters can beidentified. First, a TWE is rewritten as a system of first-order ODEs with vector field F . Thenthe condition on stationary states implies that endstates u ± and wave speed c have to satisfy f ( u + ) − f ( u − ) = c ( u + − u − ) . (30)This condition is known in shock wave theory as Rankine-Hugoniot condition (54) on theshock triple ( u − , u + ; c ). The (nonlinear) stability of hyperbolic stationary states ( u s , v s ) of F is determined by the eigenvalues λ ± = − ǫδ ± p ǫ + 4 δh ′ ( u s )2 | δ | (31)of the Jacobian D F ( u s , v s ). If ǫ, δ >
0, then ( u + ,
0) is always either a saddle or stable node,and h ′ ( u − ) = f ′ ( u − ) − c > u − ,
0) is unstable. For example, Lax’ entropycondition (55), i.e. f ′ ( u + ) < c < f ′ ( u − ), implies the latter condition. convex flux functions.Theorem 2. Suppose f ∈ C ( R ) is a strictly convex function. Let ǫ , δ be positive and let ( u − , u + ; c ) satisfy the Rankine-Hugoniot condition (54) and the entropy condition (55) , i.e. u − > u + . Then, there exists an (up to translations) unique TWS (¯ u, c ) of (26) in the sense ofDefinition 1.Proof. We consider the associated reaction-diffusion equation (29), i.e. ∂ t u = r ( u ) + δ∂ x u with r ( u ) = − h ( u ). Due to (54) and (55), r ( u ) is monostable in the sense of Definition 4. Moreover,function r is strictly concave, since r ′′ ( u ) = − f ′′ ( u ) and f ∈ C ( R ) is strictly convex. Infact, ( u ± ,
0) are the only stationary points of system (22), where ( u − ,
0) is a saddle point and( u + ,
0) is a stable node. Thus, for all wave speeds ǫ there exists a TWS (¯ u, ǫ ) – with possiblyoscillatory profile ¯ u – of reaction-diffusion equation (29). Moreover, (¯ u, c ) is a TWS of (26),due to (27)–(29). he TWP for KdVB equations (26) with Burgers’ flux f ( u ) = u has been investigatedin [12]. The sign of δ in (26) is irrelevant, since it can be changed by a transformation ˜ x = − x and ˜ u (˜ x, t ) = − u ( x, t ), see also [31]. First, the results in Theorem 2 on the existence ofTWS and geometry of its profiles are proven. More importantly, the authors investigate theconvergence of profiles ¯ u ( ξ ; ǫ, δ ) in the limits ǫ → δ →
0, as well as ǫ and δ tending tozero simultaneously. Assuming that the ratio δ/ǫ remains bounded, they show that the TWSconverge to the classical Lax shocks for this vanishing diffusive-dispersive regularization [12]. concave-convex flux functions.Definition 5 ([34]) . A function f ∈ C ( R ) is called concave-convex if uf ′′ ( u ) > ∀ u = 0 , f ′′′ (0) = 0 , lim u →±∞ f ′ ( u ) = + ∞ . (32)Here the single inflection point is shifted without loss of generality to the origin. We considera cubic flux function f ( u ) = u as the prototypical concave-convex flux function with a singleinflection point, see [29, 34]. Proposition 5 ([31, 28]) . Suppose f ( u ) = u and ǫ > .(a) If δ ≤ then a TWS (¯ u, c ) of (26) exists if and only if ( u − , u + ; c ) satisfy the Rankine-Hugoniot condition (54) and the entropy condition (55) .(b) If δ > then a TWS (¯ u, c ) of (26) exists for u − > if and only if u + ∈ S ( u − ) with S ( u − ) = ( [ − u − , u − ) if u − ≤ β , {− u − + β } ∪ [ − β, u − ) if u − > β , (33) where the coefficient β is given by β = √ ǫ √ δ .Proof. Following the discussion from (26)–(29), we consider the associated reaction-diffusionequation (29), i.e. ∂ t u = r ( u ) + δ∂ x u with r ( u ) = − h ( u ). From this point of view, we needto classify the reaction term r ( u ) = − h ( u ): Whereas r ( u − ) = 0 by definition, r ( u + ) = 0 ifand only if ( u − , u + ; c ) satisfies the Rankine-Hugoniot condition (54). The Rankine-Hugoniotcondition implies c = u + u + u − + u − . Hence, the reaction term r ( u ) has a factorization r ( u ) = − ( u − u − − c ( u − u − )) = − ( u − u − ) ( u − u + ) ( u + u + + u − ) (34)Thus, r ( u ) is a cubic polynomial with three roots u ≤ u ≤ u , such that r ( u ) = − ( u − u )( u − u )( u − u ). In case of distinct roots u < u < u we deduce r ′ ( u ) < r ′ ( u ) > r ′ ( u ) <
0. The ordering of the roots u ± and u ∗ = − u − − u + depending on u ± is visualizedin Figure 2. Next, we will discuss the results in Proposition 5(b) (for u − > δ >
0) viaresults on the existence of TWS for a reaction-diffusion equation (29).1. For u + < u ∗ < u − , function r ( u ) is bistable, see also Figure 2. Due to Proposition 4,there exists an (up to translations) unique TWS (¯ u, ǫ ) with possibly negative wave speed.Under our assumption that the wave speed ǫ is positive, relation (25) yields the restriction − u + > u − . In fact, for u − > β and u + = − u − + β there exists a TWS (¯ u, ǫ ) forreaction-diffusion equation (29), see [31, Theorem 3.4]. The function r is bistable with u ∗ = − u − − u + = − β , hence f ′ ( u ± ) > c . This violates Lax’ entropy condition (55) andis known in the shock wave theory as a slow undercompressive shock [34].2. For u ∗ < u + < u − , function r ( u ) is monostable, see Figure 2. Due to Proposition 4,there exists a critical wave speed c ∗ , such that monotone TWS (¯ u, ǫ ) for (29) exist forall ǫ ≥ c ∗ . However, not all endstates ( u − , u + ) in the subset defined by u ∗ < u + < u − admit a TWS (¯ u, c ), see (33) and Figure 3b). The TWS (¯ u, c ) associated to non-classical ∗ < u + < u − monostable − h ( u ) u + < u ∗ < u − bistable − h ( u ) u + < u − < u ∗ monostable h ( u ) monostable h ( u ) u − < u + < u ∗ bistable h ( u ) u − < u ∗ < u + monostable h ( u ) u ∗ < u − < u + u + = u − u + = − u − / ⇔ u + = u ∗ u + = − u − ⇔ u − = u ∗ u − u + Figure 2: classification of the cubic reaction function r ( u ) = − h ( u ) in (34) depending on its roots u − , u + and u ∗ = − u − − u + according to Definition 4. shocks appear again, with reversed roles for the roots u + and u ∗ : For u − > β and u + = − β , there exists a TWS (¯ u, ǫ ) for reaction-diffusion equation (29), see [31, Theorem3.4]. These TWS form a horizontal halfline in Figure 3b) and divides the set defined by u ∗ < u + < u − into two subsets. In particular, TWS exist only for endstates ( u − , u + ) inthe subset above this halfline.3. For u + < u − < u ∗ , function r ( u ) = − h ( u ) satisfies r ( u ) < u ∈ ( u + , u − ), see alsoFigure 2. Thus the necessary condition (25) can not be fulfilled for positive c = ǫ , hencethere exists no TWS (¯ u, ǫ ) for the reaction-diffusion equation.4. For u ∗ < u − < u + , function r ( u ) is monostable with reversed roles of the endstates u ± , seeFigure 2. Due to Proposition 4, there exists a TWS (¯ u, ǫ ) however satisfying lim ξ →∓∞ ¯ u ( ξ ) = u ± .If δ = 0, then equation (26) is a viscous conservation law, and its TWE (28) is a simple ODE − ǫ ¯ u ′ = r (¯ u ) with r ( u ) = − h ( u ). Thus a heteroclinic orbit exists only for monostable r ( u ), i.e.if the unstable node u − and the stable node u + are not separated by any other root of r .If δ <
0, then we rewrite TWE (28) as ǫ ¯ u ′ = h ( u ) + | δ | ¯ u ′′ . It is associated to a reaction-diffusion equation ∂ t u = h ( u ) + | δ | ∂ x u via a TWS ansatz u ( x, t ) = ¯ u ( x − ( − ǫ ) t ); note thechange of sign for the wave speed. If u + < u ∗ < u − then h ( u ) is an unstable reactionfunction. Thus there exists no TWS (¯ u, − ǫ ) according to Proposition 4. If u ∗ < u + < u − thenfunction h ( u ) = − r ( u ) satisfies h ( u ) < u ∈ ( u + , u − ), see also Figure 2. The necessarycondition (25) is still fine, since also the sign of the wave speed changed. In contrast to thecase δ >
0, there exists no TWS connecting u − with u ∗ , which would indicate a bifurcation.Thus, the existence of TWS for all pairs ( u − , u + ) in the subset defined by u ∗ < u + < u − canbe proven. The TWP for other pairs ( u − , u + ) is discussed similarly. ∗ < u + < u − monostable ru + < u ∗ < u − bistable r u + = u − u + = − u − / u − = b = √ e √ d u + = − u − u − u + (a) TWS u + = u − u − = b = √ e √ d u + = − b u + = − u − / u + = − u − + b TWS u + = − u − u − u + (b) Figure 3: a) classification of reaction function r depending on its roots u − , u + and u ∗ = − u − − u + ;b) Endstates u ± in the shaded region and on the thick line can be connected by TWS of the cubicKdVB equation; TWS in the shaded region and on the thick line are associated to classical and non-classical shocks of ∂ t u + ∂ x u = 0, respectively. For a classical shock the shock triple satisfies Lax’entropy condition f ′ ( u − ) > c > f ′ ( u + ); i.e. characteristics in the Riemann problem meet at theshock. In contrast, the non-classical shocks are of slow undercompressive type, i.e. characteristicsin the Riemann problem cross the shock. The first example of a reaction-diffusion equation with nonlocal diffusion is the integro-differentialequation ∂ t u = J ∗ u − u + r ( u ) , t > , x ∈ R , (35)for some even, non-negative function J with mass one, i.e. for all x ∈ R J ∈ C ( R ) , J ≥ , J ( x ) = J ( − x ) , Z R J ( y ) d y = 1 , (36)and some function r . The operator L [ u ] = J ∗ u − u is a L´evy operator, see (13), which modelsnonlocal diffusion. It is the infinitesimal generator of a compound Poisson stochastic process,which is a pure jump process.The TWP for given endstates u ± is to study the existence of a TWS (¯ u, c ) for (35) in thesense of Definition 1. Such a TWS (¯ u, c ) satisfies the TWE − c ¯ u ′ = J ∗ ¯ u − ¯ u + r (¯ u ) for ξ ∈ R .Next, we recall some results on the TWP for (35), which will depend crucially on the type ofreaction function r and the tail behavior of a kernel function J . We will present the existenceof TWS with monotone decreasing profiles ¯ u , which will follow from the cited literature aftera suitable transformation. roposition 6 ((monostable [21]), (bistable [11, 19])) . Suppose u − > u + and consider reactionfunctions r in the sense of Definition 4. Suppose J ∈ W , ( R ) and its continuous representativesatisfies (36) . • If r is monostable and there exists λ > such that R R J ( y ) exp( λy ) d y < ∞ then thereexists a positive constant c ∗ such that for all c ≥ c ∗ there exists a monotone TWS (¯ u, c ) of (35) . For c < c ∗ no such monotone TWS exists. • If r is bistable and R R | y | J ( y ) d y < ∞ , then there exists an (up to translations) uniquemonotone TWS (¯ u, c ) of (35) . For monostable reaction functions, the tail behavior of kernel function J is very important.There exist kernel functions J , such that TWS exist only for bistable – but not for monostable– reaction functions r , see [47]. The prime example are kernel functions J which decay moreslowly than any exponentially decaying function as | x | → ∞ in the sense that J ( x ) exp( η | x | ) →∞ as | x | → ∞ for all η > ∂ t u ( x, t ) = A [ u ( · , t )]( x ) for ( x, t ) ∈ R × (0 , T ] , where the nonlinear operator A is assumed to(a) be independent of t ;(b) generate a L ∞ semigroup;(c) be translational invariant, i.e. A satisfies for all u ∈ dom A the identity A [ u ( · + h )]( x ) = A [ u ( · )]( x + h ) ∀ x , h ∈ R . Consequently, there exists a function r : R → R which is defined by A [ υ ] = r ( υ ) for υ ∈ R and the constant function : R → R , x
1. This function r is assumed to bebistable in the sense of Definition 4;(d) satisfy a comparison principle: If ∂ t u ≥ A [ u ], ∂ t v ≤ A [ v ] and u ( · , (cid:13) v ( · , u ( · , t ) > v ( · , t ) for all t > A are needed in theproofs. Finally integro-differential evolution equations ∂ t u = ǫ∂ x u + G ( u, J ∗ S ( u ) , . . . , J n ∗ S n ( u )) (37)are considered for some diffusion constant ǫ ≥
0, smooth functions G and S k , and kernelfunctions J k ∈ C ( R ) ∩ W , ( R ) satisfying (36) where k = 1 , . . . , n . Additional assumptions onthe model parameters guarantee that an equation (37) can be interpreted as a reaction-diffusionequation with bistable reaction function including equations (20) and (35) as special cases.Another example of reaction-diffusion equations with nonlocal diffusion are the integro-differential equations ∂ t u = D aθ u + r ( u ) , t > , x ∈ R , (38)for a (particle) density u = u ( x, t ), some function r = r ( u ), and a Riesz-Feller operator D aθ with ( α, θ ) ∈ D a,θ . The nonlocal Riesz-Feller operators are models for superdiffusion, wherefrom a probabilistic view point a cloud of particle is assumed to spread faster than by followingBrownian motion. Integro-differential equation (38) can be derived as a macroscopic equation or a particle density in the limit of modified Continuous Time Random Walk (CTRW), see [37].In the applied sciences, equation (38) has found many applications, see [43, 46] for extensivereviews on modeling, formal analysis and numerical simulations.The TWP for given endstates u ± is to study the existence of a TWS (¯ u, c ) for (38) in thesense of Definition 1. Such a TWS (¯ u, c ) satisfies the TWE − c ¯ u ′ = D aθ ¯ u + r (¯ u ) , ξ ∈ R . (39)First we collect mathematical rigorous results about the TWP associated to (38) in case of thefractional Laplacian D a = − ( − ∆) a/ for a ∈ (0 , D aθ with θ = 0. Proposition 7 ((monostable [13, 14, 24]), (bistable [17, 15, 16, 39, 20, 27])) . Suppose u − >u + . Consider the TWP for reaction-diffusion equation (38) with functions r in the senseof Definition 4 and fractional Laplacian D a , i.e. symmetric Riesz-Feller operators D aθ with < a < and θ = 0 . • If r is monostable then there does not exist any TWS (¯ u, c ) of (38) . • If r is bistable then there exists an (up to translations) unique monotone TWS (¯ u, c ) of (38) . For monostable reaction functions, Cabr´e and Roquejoffre prove that a front moves expo-nentially in time [13, 14]. They note that the genuine algebraic decay of the heat kernels G a associated to fractional Laplacians is essential to prove the result, which implies that no TWSwith constant wave speed can exist. Engler [24] considered the TWP for (38) for a differ-ent class of monostable reaction functions r and non-extremal Riesz-Feller operators D aθ with( a, θ ) ∈ D + a,θ and D + a,θ := { ( a, θ ) ∈ D a,θ | | θ | < min { a, − a } } . Again the associated heatkernels G aθ ( x, t ) with ( a, θ ) ∈ D + a,θ decay algebraically in the limits x → ±∞ , see [36].To our knowledge, we established the first result [6] on existence, uniqueness (up to trans-lations) and stability of traveling wave solutions of (38) with Riesz-Feller operators D aθ for( a, θ ) ∈ D a,θ with 1 < a < r . We present our results for monotone de-creasing profiles, which can be inferred from our original result after a suitable transformation. Theorem 3 ([6]) . Suppose u − > u + , ( a, θ ) ∈ D a,θ with < a < , and r ∈ C ∞ ( R ) is a bistablereaction function. Then there exists an (up to translations) unique monotone decreasing TWS (¯ u, c ) of (38) in the sense of Definition 1. The technical details of the proof are contained in [6], whereas in [5] we give a conciseoverview of the proof strategy and visualize the results also numerically. In a forthcomingarticle [4], we extend the results to all non-trivial Riesz-Feller operators D aθ with ( a, θ ) ∈ D ⋄ a,θ .The smoothness assumption on r is convenient, but not essential. To prove Theorem 3, wefollow – up to some modifications – the approach of Chen [19]. It relies on a strict comparisonprinciple and the construction of sub- and supersolutions for any given TWS. His quantitativeassumptions on operator A are too strict, such that his results are not directly applicable. Amodification allows to cover the TWP for (38) for all Riesz-Feller operators D aθ with 1 < a < θ , and all bistable functions r regardless of the balance of the potential.Next, we quickly review different methods to study the TWP of reaction-diffusion equa-tions (38) with bistable function r and fractional Laplacian. In case of a classical reaction-diffusion equation (20), the existence of a TWS can be studied via phase-plane analysis [10, 25].This method has no obvious generalization to our TWP for (38), since its traveling wave equa-tion (39) is an integro-differential equation. The variational approach has been focused – sofar – on symmetric diffusion operators such as fractional Laplacians and on balanced poten-tials, hence covering only stationary traveling waves [39]. Independently, the same results areachieved in [17, 15, 16] by relating the stationary TWE (39) θ =0 ,c =0 via [18] to a boundaryvalue problem for a nonlinear partial differential equation. The homotopy to a simpler TWP as been used to prove the existence of TWS in case of (35), and (38) θ =0 with unbalancedpotential [27].Chmaj [20] also considers the TWP for (38) θ =0 with general bistable functions r . He ap-proximates a given fractional Laplacian by a family of operators J ǫ ∗ u − ( R J ǫ ) u such thatlim ǫ → J ǫ ∗ u − ( R J ǫ ) u = D a u in an appropriate sense. This allows him to obtain a TWSof (38) θ =0 with general bistable function r as the limit of the TWS u ǫ of (35) associated to( J ǫ ) ǫ ≥ . It might be possible to modify Chmaj’s approach to study reaction-diffusion equa-tion (38) with asymmetric Riesz-Feller operators. This would give an alternative existenceproof for TWS in Theorem 3. However, Chen’s approach allows to establish uniqueness (up totranslations) and stability of TWS as well. First we consider the integro-differential equation in multi-dimensions d ≥ ∂ t u + ∂ x f ( u ) = ǫ ∆ x u + γǫ d X j =1 (cid:0) φ ǫ ∗ ∂ x j u − ∂ x j u (cid:1) , x ∈ R d , t > , (40)for parameters ǫ > γ ∈ R , a smooth even non-negative function φ with compact support andunit mass, i.e. R R d φ ( x ) d x = 1, and the rescaled kernel function φ ǫ ( x ) = φ ( x/ǫ ) /ǫ d . It hasbeen derived as a model for phase transitions with long range interactions close to the surface,which supports planar TWS associated to undercompressive shocks of (51), see [41]. A planarTWS (¯ u, c ) is a solution u ( x, t ) = ¯ u ( x − cte ) for some fixed vector e ∈ R d , such that the profile istransported in direction e . The existence of planar TWS is proven by reducing the problem toa one-dimensional TWP for (40) d =1 , identifying the associated reaction-diffusion equation (35)and using results in Proposition 6. For cubic flux function u , the existence of planar TWSassociated to undercompressive shocks of (51) is established. Moreover, the well-posedness ofits Cauchy problem and the convergence of solutions u ǫ as ǫ ց fractal Korteweg-de Vries-Burgers equation ∂ t u + ∂ x f ( u ) = ǫ∂ x D α + u + δ∂ x u, x ∈ R , t > , (41)for some ǫ > δ ∈ R .Equation (41) with α = 1 / u ± is to study the existence of a TWS (¯ u, c ) for (41) in thesense of Definition 1. Such a TWS (¯ u, c ) satisfies the TWE h (¯ u ) := f (¯ u ) − f ( u − ) − c (¯ u − u − ) = ǫ D α + ¯ u + δ ¯ u ′′ . (42)We obtain a necessary condition for the existence of TWS – see also (25) – by multiplying theTWE with ¯ u ′ and integrating on R , Z u + u − h ( u ) d u = ǫ Z ∞−∞ ¯ u ′ D α + ¯ u ( ξ ) d ξ ≥ , (43)where the last inequality follows from (50). onnection with reaction-diffusion equation. If a TWS (¯ u, c ) for (41) exists, then u ( x, t ) =¯ u ( x ) is a stationary TWS (¯ u,
0) of the evolution equation ∂ t u = − ǫ D α + u − δ∂ x u + h ( u ) , x ∈ R , t > . (44)To interpret equation (44) as a reaction-diffusion equation, we need to verify that − ǫ D α + u − δ∂ x u is a diffusion operator, e.g. that − ǫ D α + u − δ∂ x u generates a positivity preserving semigroup. Lemma 4.
Suppose < α < and γ , γ ∈ R . The operator γ D α + u + γ ∂ x u is a L´evy operatorif and only if γ ≤ and γ ≥ . Moreover, the associated heat kernel is strictly positive if andonly if γ > .Proof. For α ∈ (0 , −D α + is a Riesz-Feller operator D α − α and generates a posi-tivity preserving convolution semigroup with a L´evy stable probability distribution G α − α as itskernel. The probability distribution is absolutely continuous with respect to Lebesgue measureand its density has support on a half-line [36]. For example the kernel associated to −D / isthe L´evy-Smirnov distribution. Thus, for γ ≤ γ ≥
0, the operator γ D α + u + γ ∂ x u isa L´evy operator, because it is a linear combination of L´evy operators. Using the notation forFourier symbols of Riesz-Feller operators, the partial Fourier transform of equation ∂ t u = −| γ |D α [ u ] + γ ∂ x u is given by ∂ t F [ u ]( k ) = ( | γ | ψ α − α ( k ) − γ k ) F [ u ]( k ). Therefore, the operator generates aconvolution semigroup with heat kernel F − [exp { ( | γ | ψ α − α ( k ) − γ k ) t } ]( x ) = G α − α ( · , | γ | t ) ∗ G ( · , γ t ) ( x ) , which is the convolution of two probability densities. The kernel is positive on R since prob-ability densities are non-negative on R and the normal distribution G is positive on R forpositive γ t .The operator D α + for α ∈ (0 ,
1) is not a Riesz-Feller operator, see Figure 1, and it generatesa semigroup which is not positivity preserving. Thus it and any linear combination with γ > convex flux functions.Proposition 8. Consider (41) with < α < , δ ∈ R and strictly convex flux function f ∈ C ( R ) . For a shock triple ( u − , u + ; c ) satisfying the Rankine-Hugoniot condition (54) , a non-constant TWS (¯ u, c ) can exist if and only if Lax’ entropy condition (55) is fulfilled, i.e. u − >u + .Proof. The Rankine-Hugoniot condition (54) ensures that h ( u ) in (42) has exactly two roots u ± .If Lax’ entropy condition (55) is fulfilled, then u − > u + and − h ( u ) is monostable in the senseof Definition 4. Thus, the necessary condition (43) is satisfied. If u − = u + then (43) impliesthat ¯ u is a constant function satisfying ¯ u ≡ u ± . If u − < u + then − h ( u ) is monostable inthe sense of Definition 4 with reversed roles of u ± . Thus, the necessary condition (43) is notsatisfied.Next, we recall some existence result which have been obtained by directly studying theTWE. In an Addendum [22], we removed an initial assumption on the solvability of the lin-earized TWE. heorem 5 ([3]) . Consider (41) with δ = 0 and convex flux function f ( u ) . For a shocktriple ( u − , u + ; c ) satisfying (54) and (55) , there exists a monotone TWS of (41) in the senseof Definition 1, whose profile ¯ u ∈ C b ( R ) is unique (up to translations) among all functions u ∈ u − + H ( −∞ , ∩ C b ( R ) . This positive existence result is consistent with the negative existence result in Proposition 7and Engler [24] for (38) with non-extremal Riesz-Feller operators D aθ for ( a, θ ) ∈ D + a,θ . Thereason is that −D α + for 0 < α < Theorem 6 ([2]) . Consider (41) with flux function f ( u ) = u / . For a shock triple ( u − , u + ; c ) satisfying (54) and (55) , there exists a TWS of (41) in the sense of Definition 1, whose profile ¯ u is unique (up to translations) among all functions u ∈ u − + H ( −∞ , ∩ C b ( R ) . If dispersion dominates diffusion then the profile of a TWS (¯ u, c ) will be oscillatory in thelimit ξ → ∞ . For a classical KdVB equation this geometry of profiles depends on the ratio ǫ /δ and the threshold can be determined explicitly. concave-convex flux functions. We consider a cubic flux function f ( u ) = u as the proto-typical concave-convex flux function. Again the necessary condition (43) and the classificationof function h ( u ) = − r ( u ) in Figure 2 can be used to identify non-admissible shock triples( u − , u + ; c ) for the TWP of (41).We conjecture that a statement analogous to Proposition 5 holds true. Of special interest isagain the occurrence of TWS (¯ u, c ) associated to non-classical shocks, which are only expectedin case of (41) with ǫ > δ > Proposition 9 (conjecture) . Suppose f ( u ) = u and ǫ > .1. If δ ≤ then a TWS (¯ u, c ) of (41) exists if and only if ( u − , u + ; c ) satisfy the Rankine-Hugoniot condition (54) and the entropy condition (55) .2. If δ > then a TWS (¯ u, c ) of (41) exists for u − > if and only if u + ∈ S ( u − ) for someset S ( u − ) similar to (33) .sketch of proof. If δ = 0, then equation (41) is a viscous conservation law, and its TWE (42)is a fractional differential equation ǫ D α + ¯ u = h (¯ u ). Thus a heteroclinic orbit exists only formonostable − h ( u ), i.e. if the unstable node u − and the stable node u + are not separated byany other root of h . This follows from Theorem 5 and its proof in [3, 22].If δ <
0, then the TWE (42) is associated to a reaction-diffusion equation (44) via astationary TWS ansatz u ( x, t ) = ¯ u ( x ). First we note that a stronger version of the necessarycondition (43) is available Z ξ −∞ h (¯ u )¯ u ′ ( y ) d y = ǫ Z ξ −∞ ¯ u ′ D α + ¯ u ( y ) d y ≥ , ∀ ξ ∈ R , (45)see [2]. If u + < u ∗ < u − then h ( u ) is an unstable reaction function, see Figure 2. Thusthere exists no TWS in the sense of Definition 1 satisfying the necessary condition (45). If u ∗ < u + < u − then function − h ( u ) is monostable in the sense of Definition 4 and the necessarycondition (43) can be satisfied. The existence of a TWS (¯ u, c ) can be proven by following theanalysis in [2, 22]. The TWP for other pairs ( u − , u + ) is discussed similarly.If δ > u, c ) associated to non-classical shocks is possible.Unlike in our previous examples, the associated evolution equation (44) is not a reaction-diffusion equation, since − ǫ D α + ¯ u − δ ¯ u ′′ is not a L´evy operator. Especially, the results onexistence of TWS for reaction-diffusion equations with bistable reaction function can not beused to prove the existence of TWS (¯ u, c ) associated to a undercompressive shocks. Instead, e investigate the TWP directly [1], extending the analysis in [2, 22] for Burgers’ flux to thecubic flux function f ( u ) = u . Fowler’s equation (8) for dune formation is a special case of the evolution equation ∂ t u + ∂ x f ( u ) = δ∂ x u − ǫ∂ x D α + u , t > , x ∈ R , (46)with 0 < α <
1, positive constant ǫ, δ > f . Here the fractional derivativeappears with the negative sign, but this instability is regularized by the second order derivative.The initial value problem for (8) is well-posed in L [7]. However, it does not support amaximum principle, which is intuitive in the context of the application due to underlyingerosions [7]. The existence of TWS of (8) – without assumptions (3) on the far-field behavior– has been proven [8].For given endstates u ± , the TWP for (46) is to study the existence of a TWS (¯ u, c ) for (46)in the sense of Definition 1. Such a TWS (¯ u, c ) satisfies the TWE h (¯ u ) := f (¯ u ) − f ( u − ) − c (¯ u − u − ) = δ ¯ u ′ − ǫ D α + ¯ u , ξ ∈ R . (47)For δ = 0, the TWE reduces to a fractional differential equation ǫ D α + ¯ u = − h (¯ u ), which hasbeen analyzed in [3, 22] for monostable functions − h ( u ).Equation (47) is also the TWE for a TWS (¯ u, δ ) of an evolution equation ∂ t u = − ǫ D α + u − h ( u ) , x ∈ R , t > . (48)For ǫ >
0, the operator is − ǫ D α + ¯ u is a Riesz-Feller operator ǫD α − α whose heat kernel G α − α has only support on a halfline. For a shock triple ( u − , u + ; c ) satisfying the Rankine-Hugoniotcondition (54), at least h ( u ± ) = 0 holds. Under these assumptions, equation (48) is a reaction-diffusion equation with a Riesz-Feller operator modeling diffusion.The abstract method in [8] does not provide any information on the far-field behavior.Thus, assume the existence of a TWS (¯ u, c ) in the sense of Definition 1, for some shock triple( u − , u + ; c ) satisfying the Rankine-Hugoniot condition (54). Again, a necessary condition isobtained by multiplying TWE (47) with ¯ u ′ and integrating on R ; hence, Z u + u − h ( u ) d u = Z R (¯ u ′ ) d ξ − Z R ¯ u ′ D α + ¯ u d ξ . (49)The left hand side is indefinite since each integral is non-negative, see also (50).For a cubic flux function f ( u ) = u and a shock triple ( u − , u + ; c ) satisfying the Rankine-Hugoniot condition (54), we deduce a bistable reaction function r ( u ) = − h ( u ) as long as u + < − u + − u − < u − see Figure 2. However, since the heat kernel has only support on ahalfline, we can not obtain a strict comparison principle as needed in Chen’s approach [19, 6, 4]. A Caputo fractional derivative on R For α >
0, the (Gerasimov-)Caputo derivatives are defined as, see [32, 43],( D α + f )( x ) = ( f ( n ) ( x ) if α = n ∈ N , n − α ) R x −∞ f ( n ) ( y )( x − y ) α − n +1 d y if n − < α < n for some n ∈ N . ( D α − f )( x ) = ( f ( n ) ( x ) if α = n ∈ N , ( − n Γ( n − α ) R ∞ x f ( n ) ( y )( y − x ) α − n +1 d y if n − < α < n for some n ∈ N . roperties: • For α > λ > D α + exp( λ · ))( x ) = λ α exp( λx ) , ( D α − exp( − λ · ))( x ) = λ α exp( − λx ) • For α > f ∈ S ( R ), a Caputo derivative is a Fourier multiplier operator with( FD α + f )( k ) = (i k ) α ( F f )( k ) where (i k ) α = exp( απ i sgn( k ) / • If ¯ u is the profile of a TWS (¯ u, c ) in the sense of Definition 1, then Z ∞−∞ ¯ u ′ ( y ) D α + ¯ u ( y ) d y = Z R ¯ u ′ ( x ) Z R ¯ u ′ ( y ) | x − y | α d y d x ≥ , (50)where the last inequality follows from [35, Theorem 9.8]. B shock wave theory for scalar conservation laws
A standard reference on the theory of conservation laws is [23], whereas [34] covers the specialtopic of non-classical shock solutions. A scalar conservation law is a partial differential equation ∂ t u + ∂ x f ( u ) = 0 , t > , x ∈ R , (51)for some flux function f : R → R . For nonlinear functions f , it is well known that the initialvalue problem (IVP) for (51) with smooth initial data may not have a classical solution forall time t > Riemann problems are a subclass of IVPs for (51), and especially important in some numericalalgorithms: For given u − , u + ∈ R , find a weak solution u ( x, t ) for the initial value problemof (51) with initial condition u ( x,
0) = ( u − , x < ,u + , x > . (52)Weak solutions of a Riemann problem that are discontinuous for t > Example . A shock wave is a discontinuous solution of the Riemann problem, u ( x, t ) = ( u − , x < ct ,u + , x > ct , (53)if the shock triple ( u − , u + ; c ) satisfies the Rankine-Hugoniot condition f ( u + ) − f ( u − ) = c ( u + − u − ) . (54)The Rankine-Hugoniot condition (54) is a necessary condition that u ± are stationary states ofan associated TWE (28), see (30). shock admissibility Classical approaches to select a unique weak solution of the Riemann problem are(a)
Lax’ entropy condition: f ′ ( u + ) < c < f ′ ( u − ) . (55)It ensures that in the method of characteristics all characteristics enter the shock/discontinuityof a shock solution (53). For convex flux function f , condition (55) reduces to u − > u + .Shocks satisfying (55) are also called Lax or classical shocks. For non-convex flux func-tions f , also non-classical shocks can arise in experiments, called slow undercompressiveshocks if f ′ ( u ± ) > c , and fast undercompressive shocks if f ′ ( u ± ) < c . b) Oleinik’s entropy condition. f ( w ) − f ( u − ) w − u − ≥ f ( u + ) − f ( u − ) u + − u − for all w between u − and u + . (56)(c) entropy solutions satisfying integral inequalities based on entropy-entropy flux pairs, suchas Kruzkov’s family of entropy-entropy flux pairs.(d) vanishing viscosity. In the classical vanishing viscosity approach, instead of (51) oneconsiders for ǫ > ∂ t u + ∂ x f ( u ) = ǫ∂ x u , t > , x ∈ R , (57)where ǫ∂ x u models diffusive effects such as friction. Equation (57) is a parabolic equation,hence the Cauchy problem has global smooth solutions u ǫ for positive times, especiallyfor Riemann data (52). An admissible weak solution of the Riemann problem is identifiedby studying the limit of u ǫ as ǫ ց R [ u ] := ǫ L [ u ] + δ∂ x L [ u ].Already for convex functions f , the convergence of solutions of the regularized equations(e.g. (1)) to solutions of (51) reveals a diverse solution structure. The solutions ofviscous conservation laws (57) converge for ǫ ց ǫ, δ → • weak dispersion δ = O ( ǫ ) for ǫ → δ = βǫ for some β > • moderate dispersion δ = o ( ǫ ) for ǫ → • strong dispersion weak limit of TWS for ǫ, δ → f , a TWS may converge to a weak solution of (51) whichis not an Kruzkov entropy solution, but a non-classical shock.A simplistic shock admissibility criterion based on the vanishing viscosity approach is theexistence of TWS for a given shock triple: Definition 6 (compare with [31]) . A solution u of the Riemann problem is called admissible (with respect to a fixed regularization R ), if there exists a TWS (¯ u, c ) in the sense of Definition 1of the regularized equation (e.g. (1)) for every shock wave with shock triple ( u − , u + ; c ) in thesolution u . References [1] F. Achleitner and C. M. Cuesta. Non-classical shocks in a non-local generalised Korteweg-de Vries-Burgers equation. work in progress.[2] F. Achleitner, C. M. Cuesta, and S. Hittmeir. Travelling waves for a non-local Korteweg–deVries–Burgers equation.
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